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The term "gyromagnetic ratio" is often used as a synonym for a different but closely related quantity, the g-factor. The g-factor, unlike the gyromagnetic ratio, is dimensionless.
For a classical rotating body
Consider a charged body rotating about an axis of symmetry. According to the laws of classical physics, it has both a magnetic dipole moment and an angular momentum due to its rotation. It can be shown that as long as its charge and mass are distributed identically (e.g., both distributed uniformly), its gyromagnetic ratio is
where q is its charge and m is its mass. The derivation of this relation is as follows:
It suffices to demonstrate this for an infinitesimally narrow circular ring within the body, as the general result follows from an integration. Suppose the ring has radius r, area mass m, charge q, and angular momentum Then the magnitude of the magnetic dipole moment is
For an isolated electron
An isolated electron has an angular momentum and a magnetic moment resulting from its spin. While an electron's spin is sometimes visualized as a literal rotation about an axis, it cannot be attributed to mass distributed identically to the charge. The above classical relation does not hold, giving the wrong result by a dimensionless factor called the electron g-factor, denoted ge (or just g when there is no risk of confusion):
Gyromagnetic factor as a consequence of relativity
Since a gyromagnetic factor equal to 2 follows from the Dirac's equation it is a frequent misconception to think that a g-factor 2 is a consequence of relativity; it is not. The factor 2 can be obtained from the linearization of both the Schrödinger equation and the relativistic Klein-Gordon equation (which leads to Dirac's). In both cases a 4-spinor is obtained and for both linearizations the g-factor is found to be equal to 2; Therefore, the factor 2 is a consequence of the wave equation dependency on the first (and not the second) derivatives with respect to space and time.
The sign of the gyromagnetic ratio, y, determines the sense of precession. While the magnetic moments shown here are oriented the same for both cases of y, the spin angular momentum are in opposite directions. Spin and magnetic moment are in the same direction for
Protons, neutrons, and many nuclei carry nuclear spin, which gives rise to a gyromagnetic ratio as above. The ratio is conventionally written in terms of the proton mass and charge, even for neutrons and for other nuclei, for the sake of simplicity and consistency. The formula is:
where is the nuclear magneton, and is the g-factor of the nucleon or nucleus in question. The ratio of equal to , is 7.622593285(47) MHz/T.
The gyromagnetic ratio of a nucleus plays a role in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). These procedures rely on the fact that bulk magnetization due to nuclear spins precess in a magnetic field at a rate called the Larmor frequency, which is simply the product of the gyromagnetic ratio with the magnetic field strength. With this phenomenon, the sign of ? determines the sense (clockwise vs counterclockwise) of precession.
Most common nuclei such as 1H and 13C have positive gyromagnetic ratios. Approximate values for some common nuclei are given in the table below.
For this reason, values of ?/ 2 ?, in units of hertz per tesla (Hz/T), are often quoted instead of ?.
The derivation of this relation is as follows: First we must prove that the torque resulting from subjecting a magnetic moment to a magnetic field is
The identity of the functional form of the stationary electric and magnetic fields has led to defining the magnitude of the magnetic dipole moment equally well as , or in the following way, imitating the moment p of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges on the two poles and vector distance between the poles under the influence of the magnetic field of earth By classical mechanics the torque on this needle is But as previously stated so the desired formula comes up. is the unit distance vector.
The model of the spinning electron we use in the derivation has an evident analogy with a gyroscope. For any rotating body the rate of change of the angular momentum equals the applied torque :
Note as an example the precession of a gyroscope. The earth's gravitational attraction applies a force or torque to the gyroscope in the vertical direction, and the angular momentum vector along the axis of the gyroscope rotates slowly about a vertical line through the pivot. In the place of the gyroscope imagine a sphere spinning around the axis and with its center on the pivot of the gyroscope, and along the axis of the gyroscope two oppositely directed vectors both originated in the center of the sphere, upwards and downwards Replace the gravity with a magnetic flux density
represents the linear velocity of the pike of the arrow along a circle whose radius is where is the angle between and the vertical. Hence the angular velocity of the rotation of the spin is
This relationship also explains an apparent contradiction between the two equivalent terms, gyromagnetic ratio versus magnetogyric ratio: whereas it is a ratio of a magnetic property (i.e. dipole moment) to a gyric (rotational, from Greek: , "turn") property (i.e. angular momentum), it is also, at the same time, a ratio between the angular precession frequency (another gyric property) and the magnetic field.
The angular precession frequency has an important physical meaning: It is the angular cyclotron frequency, the resonance frequency of an ionized plasma being under the influence of a static finite magnetic field, when we superimpose a high frequency electromagnetic field.
^"Electron gyromagnetic ratio". NIST. Note that NIST puts a positive sign on the quantity; however, to be consistent with the formulas in this article, a negative sign is put on ? here. Indeed, many references say that for an electron; for example, Weil & Bolton (2007). Electron Paramagnetic Resonance. Wiley. p. 578.[full ] Also note that the units of radians are added for clarity.